Rotation matrix to construct canonical form of a conic
If$$q(x,y)=9x^2+4xy+6y^2-10$$and$$x'=\frac1{\sqrt5}(x+2y)\ \text{and}\ y'=\frac1{\sqrt5}(-2x+y),$$then $q(x',y')=5x^2+10y^2-10$. So, $q(x',y')=0\iff\frac12x^2+y^2=1$.
Rotation matrix to construct canonical form of a conic
If$$q(x,y)=9x^2+4xy+6y^2-10$$and$$x'=\frac1{\sqrt5}(x+2y)\ \text{and}\ y'=\frac1{\sqrt5}(-2x+y),$$then $q(x',y')=5x^2+10y^2-10$. So, $q(x',y')=0\iff\frac12x^2+y^2=1$.